1. Welcome Back Homework Due next Tuesday in class (on the web…)
We’re in chapter 8…

2. Review: Large Sample Confidence Intervals
n >30 or so for means, np and n(1-p) both > 5 for proportions
Review: Large Sample Confidence Intervals
1-a confidence interval for a mean:
x +/- za/2 s/sqrt(n)
1-a confidence interval for a proportion:
p +/- za/2 p(1-p)/sqrt(n)
1-a confidence interval for the difference between two means:
x1 – x2 +/- za/2 sqrt(s21/n1+s22/n2)

3. In General: ( )
Estimate (that is normally distributed via the Central Limit Theorem)
standard deviation Za/ of estimate
+/-
This gives an interval: (Lower Bound , Upper Bound)
Interpretation: This is a plausible range for the true value of
the number that we’re estimating. a is a tuning parameter
for level of plausibility: smaller a = more conservative estimate.

4. Large Sample Confidence Intervals
np and n(1-p) > 5 for all p’s…
Large Sample Confidence Intervals
1-a confidence interval for difference between two proportions:
p1-p2 +/- za/2 sqrt[(p1(1-p1)/n1)+(p2(1-p2)/n2)]

5. Designing an Experiment and Choosing a Sample Size
Example: Compare the shrinkage in a tumor due to a “new” cancer treatment relative to standard treatment
100 patients randomly assigned to “new” treatment or standard treatment
xinew = reduction in tumor size for person i under new treatment
xjstd = reduction in tumor size for person j under std treatment
xnew and s2new
xstd and s2std
Mean and sample variance of the changes in size for the new and standard treatments

6. Suppose the data are: What can we conclude? xnew = 25.3 snew = 2.0
xstd = 24.8
sstd = 2.3
95% Confidence Interval for difference:
x1 – x2 +/- za/2 sqrt(s21/n1+s22/n2)
= 0.5 +/- 0.84
What can we conclude?

7. There’s no difference?
Can’t see a difference?
There’s a difference, but it’s too small to care about?

8. There is a difference between:
Can’t see a difference
There’s no difference
Situation for
Cancer example
(In cancer experiment, we can assume we care about small differences.)

9. Can’t see a difference (that is big enough to care about) = wasted experiment
AVOID / PREVENT THE WASTE AND ASSOCIATED TEARS USE SAMPLE SIZE PLANNING

10. Sample Size Planning
Length of a 1-a level confidence interval is: “2 za/2 std deviation of estimate”
2za/2s/sqrt(n)
2za/2p(1-p)/sqrt(n)
2za/2sqrt((s21/n1)+(s22/n2))
2za/2sqrt[(p1(1-p1)/n1)+(p2(1-p2)/n2)]

11. Suppose we want a 95% confidence interval no wider than W units.
a is fixed. Assume a value for the standard deviation (or variance) of the estimator.
Solve for an n (or n1 and n2) so that the width is less than W units.
When there are two sample sizes (n1 and n2), we often assume that n1 = n2.

12. Cancer example
Let W = 0.1. Want 95% CI for difference between means with width less than W.
Suppose s2new = s2std = 6 (conservative guess)
W > 2za/2sqrt((s2new/n1)+(s2std/n2))
0.1 > 2(1.96)sqrt(6/n + 6/n)
0.1 > 3.92sqrt(12/n)
0.01 > (3.922)12/n
n > (each group…)
Book’s B = (our W)/2